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Talk:Friedman's circle theorem
The moment I saw the title my stomach dropped. Then I read the article and sighed in relief, for it was only epsilon zero. Will look over this later and try to explain this to other laymen like me. A diagram is definitely needed. FB100Z • talk • 15:26, April 21, 2013 (UTC) You can take a look at Friedman's paper ("Enormous integers in real life") to see extension of that concept to p-circles - unions of bounded number non-necessarily disjoint circles. This one reaches small Veblen ordinal! LittlePeng9 (talk) 15:42, April 21, 2013 (UTC) :WHAT. FB100Z • talk • 17:32, April 21, 2013 (UTC) :Unfortunately, Friedman also doesn't give laymen explanation, so it doesn't helps. I believe that someone should write more informally about this subject, as it has been done for busy beaver function. Ikosarakt1 (talk ^ ) 18:55, April 21, 2013 (UTC) :Do you think we should make second article for p-circles theorem, or just add another paragraph about it? In either case, I can write about it. LittlePeng9 (talk) 19:00, April 21, 2013 (UTC) :Probably it will be helpful to describe what homeomorphism means in more easy-to-understand sense. Ikosarakt1 (talk ^ ) 19:25, April 21, 2013 (UTC) I think that the inverse of a continuous function is always continuous. So a homeomorphism is just a continuous bijection. Also, I think the theorem is unprovable in Peano arithmetic. FB100Z • talk • 17:56, April 21, 2013 (UTC) Here are examples of continuous but not bicontinuous functions. For example, every continuous bijective mapping \(\mathbb{R} \to \mathbb{S}^1\) (real line to unit circle) musn't have continuous inverse. I believe there are bijections of plane with same property. LittlePeng9 (talk) 18:38, April 21, 2013 (UTC) :Hm. My intuition was wrong then. FB100Z • talk • 19:25, April 21, 2013 (UTC) Sources? Jiawhein \(a\)\(l\) 11:02, April 23, 2013 (UTC) :Added. LittlePeng9 (talk) 11:15, April 23, 2013 (UTC) I was thinking about p-circle theorem, and I found that there is probably no layman explantation of this problem. We at least would need to say when two p-circles for same p are homeomorphic, and I found no obvious way to tell that. I also found that there are 5 distinct 2-circles, and lots of distinct 3-circles, I certainly haven't found all. LittlePeng9 (talk) 19:35, May 10, 2013 (UTC) Edit: 6 2-circles. 2 identical overlapping circles are 2-circle too. LittlePeng9 (talk) 19:49, May 10, 2013 (UTC) :Hmmm, two circles can be adjacent, one inside the other, kissing, intersecting, or overlapping. What's the sixth possibility? Deedlit11 (talk) 06:30, May 12, 2013 (UTC) Internaly tangent circles. LittlePeng9 (talk) 07:01, May 12, 2013 (UTC) Homeomorphism I'm missing something here. Since \(i \neq j\), in general \(C_{i,2i}\) is a different length from \(C_{j,2j}\). How can there be a homeomorphism between the two sequences of circles? How do you continuously map 6 circles onto 5? FB100Z • talk • 21:04, May 10, 2013 (UTC) The homeomorphism has whole plane as domain. C_i is embedded to C_j, which means that restriction of function is onto. C_i doesn't have to be 1-1 with C_j. LittlePeng9 (talk) 21:27, May 10, 2013 (UTC) :I don't understand the second sentence. FB100Z • talk • 04:23, May 11, 2013 (UTC) :The point is that map (fancy name for function \(\mathbb{R}^n\rightarrow\mathbb{R}^n\) taking C_i to C_j is only required to be injective. It only needs to map i circles from C_i to i circles from C_j. LittlePeng9 (talk) 09:26, May 11, 2013 (UTC) ::Ah, now that makes more sense. I'll work on getting a more intuitive feel for this theorem and trying to explain it on the page. FB100Z • talk • 18:22, May 11, 2013 (UTC) Note that while you can continuously map 6 circles onto 5, there's no way to have a homeomorphism of the plane that maps 6 circles onto 5 - the number of circles can't decrease. Deedlit11 (talk) 00:22, May 11, 2013 (UTC) :Yes, I meant "homeomorphically map." FB100Z • talk • 04:23, May 11, 2013 (UTC) Question Why can't Friedman's circle theorem be stated in the language of arithmetic? -- 07:26, September 30, 2015 (UTC) :Because if we can only speak of natural numbers (and not sets of these) like in Peano arithmetic, then we can't even speak of points in the plane, let alone homeomorphisms. LittlePeng9 (talk) 09:53, September 30, 2015 (UTC)